FINANCIAL CRISIS:
The following formula calculates the compound interest and is used by
the banks. The formula is so designed to
calculate the interest n times the year and each time the interest is added to
the principle enhancing it continuously.
This is called the compounding the interest.
S = P*(1+r//n)^n*t
S--- Amount yielded.
P --- Principle
r--- Percentage rate of interest
n ---number of times interest compounded in a year.
t---- Number of years.
First, let us take the principle P as one dollar, rate of interest r as
100% = 100/100= 1; time t as one year and number of times n = 1, the interest
is calculated only once at the end of the year.
So what is the yield S?
S=1*(1+1/1)^1*1=1*(1+1)^1=2
So one dollar becomes 2 dollar at the
end of a year for 100 % rate of interest.
Now let us compound the interest 3 times the year- i.e.- every four
months. So n = 3, retaining other
quantities same, the yield will be
S=1*(1+1/3)^1*3=1*(1+1/3)^3=2.37
$1 becomes 2.37 dollar in a year,
when the interest is compounded every 4 months.
Now, let us do it 6 times a year- i.e.- every 2 months.
S=1*(1+1/6)^1*6=1*(1+1/6)^6=2.52
So 1 dollar yields 2.52 dollars, when the interest is compounded every 2
months
Suppose we compound the interest every day, every hour, or every second,
will the yield sky rocket? Let us find
out
No.of times
a year-compounded. n
|
Amount yielded
P
|
1
2
3
5
10
100
1000
10000
100000
1000000
.
.
.
|
2
2.25
2.37
2.488
2.5937
2.7048
2.7169
2.71814
2.718268
2.7182804.
.
.
.
|
So the yield reaches a dead end.
The one dollar cannot be grown more than the magic number 2.7182804....,
however the larger n may be. Here
we try to compound (grow) the interest as continuously as
possible, but one dollar only becomes
2.71.. dollar in one year in the seamless continuous growth.
NATURES IDEA: – A CONSTANT FOR GROWTH
The limit 2.7182804...
is known as euler's constant and is denoted by ” e”. An important
mathematical constant on par with π.
DEFINITION an attempt
' The constant e = 2.7182804 … is the maximum yield when unit quantity grows continuously at unit rate in unit time'
The above definition holds good for decay process also. We can simply say e is the unit of
natural growth or decay.
Many natural processes of growth and decay does not take place step by
step, but rather continuously. Wherever
continuous change takes place, e will show its face in some form.
Examples: growth of population, growth of a baby, rise
and fall of electric current, fall of temperature of hot cup of coffee, etc..